"Hij slaapt nadat hij heeft gedronken."

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I put 'he sleeps after he drank' and it told me explicitly I need to put 'he has drunk'. Those two things are basically equivalent in this case, aren't they? (I'm an English speaker.) Is there a nuance of the Dutch that I'm not getting?

"He sleeps" in a general sense "after he has drunk" whenever he drinks/whenever he has drunk. This sentence doesn't work if you change it to "after he drank" because then the first part needs to be in the past too: "He slept after he drank".

"Nadat" is a subordinating conjunction, used when connecting a subordinate clause to the main clause:
"Nadat ik heb ontbijt gegeten, fiets ik naar werk."
"After I have eaten breakfast, I bike to work."

"Na" is typically followed by a noun and is not used to separate clauses:
"Na het ontbijt, heb ik naar mijn werk gefiets."
"After breakfast, I biked to work."